Handling task prioritization as a product of Gaussians · Representing task prioritization as a...

Handling task prioritization as a product of Gaussians

Sylvain CalinonSenior ResearcherIdiap Research Institute, Martigny, Switzerland

LecturerEPFL, Lausanne, Switzerland

Research Groups:• Speech & Audio Processing• Natural Language Understanding• Perception & Activity Understanding• Machine Learning• Social Computing• Biometrics Security and Privacy• Biosignal Processing• Computational Bioimaging• Energy Informatics • Uncertainty Quantification and Optimal Design• Robot Learning & Interaction

Artificial Intelligence for Society

MARTIGNY

ResearchEducation

Technology transfer

Learning from demonstration Observational learning

Kinesthetic teachingCorrespondence problems

Superposition Fusion

t=0 t=1 t=2t=0 t=1 t=2

Motivating example:

A probabilistic view on segment crossing!

Many algorithms in robotics can be re-interpreted as products of Gaussians, including task prioritization!

(superposition)

(fusion)

center of the Gaussian

covariance matrix

precision matrix

Combination of primitives as a fusion problem

Scalar superposition as a product of Gaussians with :

Combination of primitives as a fusion problem

The full weight matrices approach covers both scalar weights

(with isotropic diagonal matrix) and null space projection

operations!

Null space projection(hierarchy constraints)

Scalar superposition

Tasks prioritization as PoGPrincipal task:track horizontal reference

Secondary task:track desired posture

Principal task

Secondary task

Principal task

Secondary task

Learning tasks prioritization

Parallel organization of motion/skill primitives

Task-parameterized Gaussian mixture model (TP-GMM)

[Canal, Pignat, Alenya, Calinon and Torras, ICRA’2018]

Calinon, S. (2016). A Tutorial on Task-Parameterized Movement Learning and Retrieval. Intelligent Service Robotics (Springer), 9:1, 1-29.

Task-parameterized Gaussian mixture model (TP-GMM)

[Calinon, Alizadeh and Caldwell, IROS’2013]

Coordinatesystem as task

parameter

Task-parameterized Gaussian mixture model (TP-GMM)

Candidate hierarchy

Candidate hierarchy

Demonstration

Reproduction

Demonstration

Reproduction

[Silvério, Calinon, Rozo and Caldwell, IEEE T-RO 2019]

Priority on left hand

Learning tasks prioritization

Dr João Silvério

Candidate hierarchy

Candidate hierarchy

Demonstration

Reproduction

Demonstration

Reproduction

Learning tasks prioritization

[Silvério, Calinon, Rozo and Caldwell, IEEE T-RO 2019]

Priority on right hand

Dr João Silvério

Candidate hierarchy

Candidate hierarchy

Demonstration

Reproduction

Demonstration

Reproduction

Learning tasks prioritization

[Silvério, Calinon, Rozo and Caldwell, IEEE T-RO 2019]

Equal priority

Dr João Silvério

Learning tasks prioritization

[Silvério, Calinon, Rozo and Caldwell, IEEE T-RO 2019]

Centaurorobot

Dr João Silvériobase position > end-effector positions > end-effector orientations

Learning tasks prioritization

[Silvério, Calinon, Rozo and Caldwell, IEEE T-RO 2019]

Centaurorobot

Dr João Silvérioend-effector positions > base position > end-effector orientations

Ongoing work

Learning tasks prioritization

Expert A>B>CExpert A>C>BExpert B>A>CExpert B>C>AExpert C>A>BExpert C>B>A

Expert A>BExpert B>AExpert B>CExpert C>B

Geodesic interpolation between nullspace structures

Representing task prioritization as a product of Gaussians provides a way to geometrically interpolate task hierarchies (geodesics on symmetric positive semidefinite manifolds)

Task 1 Task 2

Task 1 > Task 2 Task 2 > Task 1

Task 1 Task 1 Task 2Task 2

Exploiting redundancy in planning tasks

[Girgin and Calinon, arXiv:1905.09679, 2019]Hakan Girgin

Products of Uni-Gauss experts

Uni-Gauss experts are distributions combining the task constraint in the form of a Gaussianwith a uniform distribution

Intl Conf. on Artificial Neural Networks (ICANN’99)

Product of Gaussians

Product of Uni-Gauss experts

with the same πi

(no hierarchy; simply go from OR to AND)

Product of Uni-Gauss experts

with different πi

(task 1 is set as being obligatory)

Products of Uni-Gauss experts

Instead of providing a solution as an exact compromise between objectives (which might fulfill none of them), Uni-Gauss experts allow to represent tasks in which we might prefer to fulfill one or another, but not both.

Talos robot

Products of Uni-Gauss expertsVariational inference can be used to create distributions of good and diversifiedconfigurations.

Combined with Uni-Gauss experts, the system can sample robot configurations satisfying constraints.

[Pignat, Lembono and Calinon, arXiv:1905.09597, 2019]

Emmanuel Pignat

Conclusion

A product of Gaussians can be used to represent nullspace projection operations and weighted superposition of tasks (fusion problem)

Learning task prioritization can be achieved by a statistical analysis of the data transformed with different candidate projections (TP-GMM)

Demo

Repro

Variational inference with Uni-Gauss distributions is a promising approach to learn task constraints individually, and then combine them with prioritization

Nullspace structures are present in a wide range of robotics problems, including planning problems computed with model predictive control (MPC)

Robot Learning & Interaction Group at Idiap:

Contact:

sylvain.calinon@idiap.chhttp://calinon.ch

Source codes (Matlab/Octave, C++ and Python):

http://www.idiap.ch/software/pbdlib/

Photo: Basilio Noris

Thibaut Kulak

Emmanuel Pignat

NoémieJaquier

Dr Antonio Paolillo

HakanGirgin

MartinTroussard

TeguhLembono

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